Improper Integrals

Improper Integrals

In defining the Riemann integral _a∫^b f ( x)dx , the interval [a , b] of integration is finite and f ( x) is finite at every point in [a , b] . In many physical applications, the following types of integrals arise:

_a∫^∞ f ( x) dx , ^a∫_{− ∞} f ( x) dx , ^∞∫_{−∞} f ( x) dx ,

where a is a real number and f ( x) is a continuous function on the interval of integration. They are defined as the limits of Riemann integrals as follows:

They are called improper integrals of first kind. If the limits exist, then the improper integrals are said to be convergent.

Note

By the Fundamental theorem of integral calculus, there exists a function F (t) such that